Port Hamiltonian Formulation of Infinite Dimensional Systems I. Modeling

Transcription

1 Port Hamiltonian Formulation of Infinit Dimnsional Systms I. Modling Alssandro Macchlli, Arjan J. van dr Schaft and Claudio Mlchiorri Abstract In this papr, som nw rsults concrning th modling of distributd paramtr systms in port Hamiltonian form ar prsntd. h classical finit dimnsional port Hamiltonian formulation of a dynamical systm is gnralizd in ordr to cop with th distributd paramtr and multivariabl cas. h rsulting class of infinit dimnsional systms is quit gnral, thus allowing th dscription of svral physical phnomna, such as hat conduction, pizolctricity and lasticity. Furthrmor, classical PDEs can b rwrittn within this framwork. h ky point is th gnralization of th notion of finit dimnsional Dirac structur in ordr to dal with an infinit dimnsional spac of powr variabls. In this way, also in th distributd paramtr cas, th variation of total nrgy within th spatial domain of th systm can b rlatd to th powr flow through th boundary. Sinc this rlation dply rlis on th Stoks thorm, ths structurs ar calld Stoks Dirac structurs I. INRODUCION Following th sam idas bhind th bond graph formalism 11], a finit dimnsional physical systm can b modld as th rsult of th intrconnction of a small st of atomic lmnts, ach of thm charactrizd by a particular nrgtic bhavior.g. nrgy storing, dissipation or convrsion. Each lmnt can intract with th nvironmnt by mans of a port, that is a coupl of input and output signals whos combination givs th powr flow. h ntwork structur allows a powr xchang btwn ths componnts and dscribs th powr flows within th systm and btwn th systm and th nvironmnt. his ntwork can b mathmatically dscribd by mans of a Dirac structur 1], ], 7], 14], gnralization of th wllknown Kirchoff laws of circuit thory, 8]. Onc th Dirac structur is dfind, th dynamics of th systm is spcifid whn th spac of nrgy stat variabls and th nrgy Hamiltonian function ar givn. h port Hamiltonian formalism 7], 14] is basd on ths idas and allows th dscription of a wid class of finit dimnsional non-linar systms, such as mchanical, lctro-mchanical, hydraulic and chmical ons. h port Hamiltonian rprsntation of a finit dimnsional systm has bn rcntly xtndd in ordr to cop with th infinit dimnsional cas, 15], thus gnralizing his work has bn don in th contxt of th Europan sponsord projct GoPlx, rfrnc cod IS Furthr information is availabl at A. Macchlli and C. Mlchiorri ar with CASY DEIS, Univrsity of Bologna, vial Risorgimnto, 4136 Bologna, Italy {amacchlli, cmlchiorri}@dis.unibo.it A. J. van dr Schaft is with th Dpartmnt of Applid Mathmatics, Univrsity of wnt, 75 AE Enschd, h Nthrlands a.j.vandrschaft@math.utwnt.nl th classical Hamiltonian formulation of a distributd paramtr systm which is a wll-stablishd mathmatical rsult, 1], 13]. From th ntwork modling prspctiv, th dynamics of an infinit dimnsional systm with spatial domain and boundary is th rsult of th intraction among at last two nrgy domains within and/or btwn th systm and its nvironmnt through. his intraction is mathmatically dscribd by a gnralization of th Dirac structur to th distributd paramtr cas. Sinc this nw class of powr consrving intrconnction dply rlis on th Stoks thorm, w spak about Stoks Dirac structur. In 15], a simpl Stoks Dirac structur has bn introducd and it has bn shown that it is can b th starting point for th dscription in port Hamiltonian form of th tlgraphr quation, of Maxwll s quations and of th vibrating string quation. Morovr, in 9], this Stoks- Dirac structur has bn modifid in ordr to modl fluid dynamical systms and in 3], 6] to modl th imoshnko bam quation. In any cas, it is not compltly clar how a gnral formulation of a multi-variabl distributd paramtr systm within th port Hamiltonian formalism could b obtaind. In this papr, som nw rsults in this dirction ar prsntd. In particular, a novl class of Dirac structurs ovr an infinit dimnsional spac of powr variabls ar introducd. h intrconnction, damping and input/output matrics ar rplacd by matrix diffrntial oprators which ar assumd to b constant, that is no xplicit dpndnc on th stat nrgy variabls is considrd. As in finit dimnsions, givn th Stoks Dirac structur, th modl of th systm asily follows onc th Hamiltonian function is spcifid. h rsulting class of infinit dimnsional systms in port Hamiltonian form is quit gnral, thus allowing th intrprtation of classical PDEs within this framwork and th dscription of svral physical phnomna, as th hat conduction, pizo lctricity and lasticity. his work is organizd as follows: aftr a short background about finit dimnsional Dirac structurs and port Hamiltonian systm in Sct. II, th infinit dimnsional Stoks Dirac structurs ar introducd in Sct. III and th corrsponding port Hamiltonian formulation of multivariabl infinit dimnsional systm mdph systms is discussd in Sct. IV. In Sct. V, som simpl xampls ar prsntd, th Harry Dym quation, a classical nonlinar PDE, and th hat quation. Finally, conclusions ar discussd in Sct. VI.

2 II. DIRAC SRUCURES AND FINIE DIMENSIONAL POR HAMILONIAN SYSEMS A. Background on Dirac structurs h intrconnction of physical systm basically is powr xchang. In ordr to mathmatically modl ths phnomna, it is ncssary to giv a dfinition of powr and to introduc a propr st of tools that will b usful to trat and dscrib th ntwork structur bhind a physical systm. Considr an n-dimnsional linar spac F and dnot by E F its dual, that is th spac of linar oprator : F R. h lmnts blonging to F ar calld flows.g. vlocitis and currnts, whil th lmnts in E ar calld fforts i.. forcs and voltags. Flows and fforts ar th port variabls, that is th input and output signals, whos combination givs th powr flowing insid th physical systm. h spac F E is calld spac of powr variabls. Givn an ffort E and a flow f F, dfin th associatd powr P as P :=, f = f R whr, is th dual product btwn f and. Basd on th dual product, th following linar oprator is wlldfind. Dfinition.1 +pairing oprator: Considr th spac of powr variabls F E. h following symmtric bilinar form is wll-dfind: f 1, 1, f, := 1, f +, f 1 1 with f i, i F E, i = 1, ;, is calld +pairing oprator. Considr a linar subspac S F E of dimnsion m and dnot by S its orthogonal complmnt with rspct to th +pairing oprator 1, which is again a linar subspac of F E with dimnsion nm sinc 1 is a non-dgnrat form. Basd on th +pairing oprator 1, it is possibl to giv th fundamntal dfinition of Dirac structur, that is th basic mathmatical tool that is usd to dscrib th intrconnction structur btwn physical systms. Dfinition. Dirac structur: Considr th spac of powr variabls F E and th symmtric bilinar form 1. A constant Dirac structur on F is a linar subspac D F E such that D = D Not.1: It is possibl to prov that th dimnsion of a Dirac structur D on an n-dimnsional spac F is qual to n. his rsult is rlatd to an intrsting proprty of physical systms. Considr, for xampl, th intrconnction of lctrical ntworks: it is wll known that it is not possibl to impos both currnts and voltags. By gnralization, a physical intrconnction cannot dtrmin both th flow ithr th ffort. Morovr, suppos that f, D; from 1, w hav that = f,, f, =, f hn, it can b dducd that, for vry f, D,, f = or, quivalntly, that vry Dirac structur D on F dfins a powr-consrving rlation btwn powr variabls f, F E. With th following proposition, a quit gnral class of Dirac structurs is introducd, 14]. Proposition.1: Considr th spac of powr variabls F E and dnot by X an n-dimnsional spac, th spac of nrgy variabls. Suppos that F := F s, F r, F and that E := E s, E r, E, with dim F s = dim E s = n, dim F r = dim E r = n r and dim F = dim E = m. Morovr, dnot by Jx a skw-symmtric matrix of dimnsion n and by G r x and Gx two matrics of dimnsion n r n and m n rspctivly. hn, th st D := { f s, f r, f, s, r, F E f s = Jx s G r xf r Gxf r = G r x s = G x s } is a Dirac structur on F Not.: In Df.., th pairs f s, s and f r, r rprsnt th port variabls of th storag and dissipativ lmnts rspctivly, whil f, ar th port variabls through which th nvironmnt can xchang powr with th systm. Givn th intrconnction structur, th dynamics of th systm can b spcifid onc th port bhavior of th nrgy storag lmnts is spcifid and whn th dissipativ ports ar trminatd. B. Finit dimnsional port Hamiltonian systms h Dirac structur introducd in Df.. is quit gnral. Basd on that, a gnral formulation of non-linar systm in port Hamiltonian form can b asily givn. As discussd in Not., a dynamical systm can b intrprtd as th rsult of th combination of th Dirac structur with th port bhavior of th nrgy storing and of th dissipativ lmnts. Undr th sam hypothsis of Prop..1, dnot by H : X R a ral valud function boundd from blow dfind ovr th spac of nrgy variabls X. hn, dfin th port bhavior of th nrgy storing lmnts as follows: f s = ẋ s = H whr th minus sign is ncssary in ordr to hav a consistncy in th powr flow. If rstrictd to th linar cas, dissipativ ffcts can b takn into account by imposing th following rlation on th variabls f r, r of th Dirac structur : f r = Y r r 4 whr Y r = Yr. By substitution of 3 and 4 in, th rprsntation of a port Hamiltonian systm with 3

3 dissipation can b dducd 7], 14] and th following dfinition maks sns. Dfinition.3 finit dim. port Ham. systms: Dnot by X an n-dimnsional spac of stat nrgy variabls and by H : X R a scalar nrgy function Hamiltonian boundd from blow. Dnot by U F an m-dimnsional linar spac of input variabls and by its dual Y E th spac of output variabls. hn, ẋ = Jx Rx] H + Gxu y = G x H 5 with Jx = J x, Rx = R x and Gx matrics of propr dimnsions, is a port Hamiltonian systm with dissipation. h n n matrics J and R ar calld intrconnction and damping matrix rspctivly. Not.3: Givn a dynamical systm in port Hamiltonian from 5, th variation of intrnal nrgy quals th dissipatd powr plus th powr providd to th systm by th nvironmnt, that is: dh dt = H Rx H + y u y u his rlation xprsss a fundamntal proprty of port Hamiltonian systms, thir passivity. Roughly spaking, th intrnal nrgy of th unforcd systm u = is nonincrasing along systm trajctoris or, if th port variabl ar closd on a dissipativ lmnt, that is a rlation similar to 4 is imposd btwn u and y, thn th nrgy function is always a dcrasing function. If th dfinition of Lyapunov stability is rcalld, togthr with th sufficint condition for th stability of an quilibrium point, thn it can b dducd that th Hamiltonian is a good candidat for bing a Lyapunov function. III. POWER CONSERVING INERCONNECIONS IN INFINIE DIMENSIONS A. Constant matrix diffrntial oprators In th finit dimnsional formulation 5 of a port Hamiltonian systm, an important rol is playd by th intrconnction, damping and input matrics. hs oprators ar strictly rlatd to th proprtis of th Dirac structur dfining th powr flows within th dynamical systm and btwn th systm and its nvironmnt. In infinit dimnsions, ths objcts ar gnralizd and thy ar mathmatically dscribd by matrix diffrntial oprators. In this papr, only th constant cas is takn into account. In th finit dimnsional framwork, this mans that th dpndnc on th x variabl of th lmnts of th Dirac structur is nglctd. Dnot by a compact subst of R d rprsnting th spatial domain of th distributd paramtr systm. hn, dnot by U and V two sts of smooth functions from to R qu and R qv rspctivly. Dfinition 3.1 constant matrix diffrntial oprator: A constant matrix diffrntial oprator of ordr N is a map L from U to V such that, givn u = u 1,..., u qu U and v = v 1,..., v qv V N v = Lu v b := Pa,bD α α u a 6 #α= whr α := {α 1,..., α d } is a multi-indx of ordr #α := d i=1 α i, P α ar a st of constant q u q v matrics and D α := α1 1 is an oprator rsulting from a α d z d combination of spatial drivativs. Not that, in 6, th sum is intndd ovr all th possibl multi-indxs α with ordr to N and, implicitly, on a from 1 to q. Dfinition 3. formal adjoint: Considr th constant matrix diffrntial oprator 6. Its formal adjoint is th map L from V to U such that u = L v u b := N 1 #α Pb,aD α α v a 7 #α= Dfinition 3.3 skw-adjoint matrix diff. op.: Dnot by J a constant matrix diffrntial oprator. hn, J is skwadjoint if and only if J = J Not 3.1: It is asy to prov that, L is a skw-adjoint matrix diffrntial oprator if and only if P α a,b = 1 #α P α b,a for vry multi-indx α from ordr to N. An important rlation btwn a diffrntial oprator and its adjoint is xprssd by th following lmma, which gnralizs an analogous rsult prsntd in 1] to th multi variabl cas. As it will b discussd in Sct. III- B, this rsult is fundamntal in th dfinition of Stoks Dirac structur and, basically, it gnralizs th wll-known intgration by parts formula. Lmma 3.1: Considr a matrix diffrntial oprator L and dnot by L its formal adjoint. hn, for vry functions u U and v V, w hav that v Lu u L v ] dv = B L u, v da 8 whr B L is a diffrntial oprator inducd on by L. Not 3.: Givn u U and v V, from th Stoks thorm, it is wll known that rlation 8 can b quivalntly writtn as v Lu u L v = div B L u, v that is v Luu L v can b xprssd in divrgnc form. From 6 and 7, w hav that v Lu u L v = N = Pa,b α D α u a v b 1 #α D α v b u a] 9 #α= whos divrgnc form is #β=1 D β N 1 #γ Pa,b α D γ v b D αβγ u a 1 α β γ αβ

4 in which th first sum is xtndd to all th multi indx β of ordr 1. Not 3.3: It is important to not that B L is a constant diffrntial oprator. h quantity B L u, v is a constant linar combination of th functions u and v togthr with thir spatial drivativs up to a crtain ordr and dpnding on L. Consquntly, dnot by B an oprator providing a vctor with all th spatial drivativs in 1 and by BL i, i = 1,..., d, a st of constant squar matrics of a crtain ordr givn by a propr combinations of all th P α matrics. hn, Bu BLB 1 v BLB d v ] da givs th intgral ovr in 8. Corollary 3.: Considr a skw-adjoint matrix diffrntial oprator J. hn, for vry functions u U and v V with q u = q v, w hav that v Ju + u Jv ] dv = B J u, v da 11 whr B J is a non-dgnrat symmtric diffrntial oprator on dpnding on th diffrntial oprator J. Proof: It is immdiat from Df. 3.3 and th prvious lmma. Not 3.4: From Not 3.3, th intgral on in 11 can b altrnativly writtn as Bu BJB 1 v BJB d v ] da B. Constant Stoks Dirac structurs As in finit dimnsions, th dfinition of a powr consrving intrconnction structur is possibl onc th notion of powr is proprly introducd. Dnot by F th spac of flows and assum that F is th spac of smooth functions from th compact st R d to R q. As far as concrns th spac of fforts E, assum for simplicity that E F. hn, givn f = f 1,..., f q F and = 1,..., q E, dfin th dual product as follows: q, f := i f i dv = f dv i=1 From Df..1, th +pairing oprator on F E is givn by f 1, 1, f, := 1 f + ] f 1 dv whr f 1, 1, f, F E. Dnot by J a skw-adjoint diffrntial oprator and considr th following subst of th spac of powr variabls: D := { f, F E f = J } 1 hn, for vry f i, i D, i = 1,, w hav that f 1, 1, f, = 1 f + ] f 1 dv = 1 J + ] J 1 dv = B J 1, da 13 If only th lmnts of D with compact support on ar considrd, thn th rsulting subst of F E is a Stoks Dirac structur on F, as it can b dirctly dducd from Df.. sinc th intgral ovr is qual to. In gnral, whn an xchang of powr btwn systm and nvironmnt through th boundary of th spatial domain is prsnt, 1 is not a Stoks Dirac structur bcaus also th boundary trms hav to b takn into account. hs boundary trms ar th rstriction of th fforts and thir spatial drivativs on. Dnot by w := B th boundary trms, whr B th oprator providing th rstriction on of th ffort and of its spatial drivativs of propr ordr as discussd in Not 3.3 and Not 3.4. In this way, it is possibl to writ with som abus in notation: B J 1, da = B 1 J w BJw d ] da w1 with w i = B i, i = 1, and whr, in th last intgral, B j J, j = 1,..., d, ar th squar constant matrics introducd in Not 3.4. Furthrmor, basd on B, th following st rprsnting th spac of boundary conditions can b introducd: W := {w w = B, E} 14 hn, th following proposition can b provd. Proposition 3.3: Considr th xtndd spac of powr variabls F E W and dnot by J a skw-adjoint diffrntial oprator. hn, th following subst D J := { f,, w F E W f = J, w = B } 15 is a Stoks Dirac structur on F with rspct to th pairing f 1, 1, w 1, f,, w J := := 1 f + ] f 1 dv + B J w 1 w da 16 Proof: h proof is immdiat from 13 and 14. Not 3.5: From th proprtis of a Stoks Dirac structur, summarizd in Not.1 for th finit dimnsional cas, if f,, w D, thn f,, w, f,, w J =, that is f dv = 1 w B 1 Jw BJw d ] da his rlation, bsid xprssing th powr consrvation proprty of th Stoks Dirac structur, is abl to rlat th variation of intrnal nrgy th intgral on th spatial domain with th powr flowing insid th domain through th boundary th intgral on. h Stoks Dirac structur introducd in Prop. 3.3 is dvlopd around a skw-adjoint diffrntial oprator which inducs a non-dgnrat diffrntial oprator on th boundary. In finit dimnsions, this situation can b obtaind by assuming G r = G = in th Dirac structur of Prop..1, that is by assuming that th powr consrving ntwork intrconncts only a st of nrgy storing lmnts. It is intrsting to compltly gnraliz th rsult of Prop..1

5 to th distributd paramtr cas or, quivalntly, to proprly modify th Stoks Dirac structur 15 of Prop. 3.3 in ordr to tak into account dissipativ ffcts and an intraction btwn systm and nvironmnt along th spatial domain and not only through th boundary. h last situation can b ncountrd, for xampl, in th cas of Maxwll s quations whn a currnt dnsity diffrnt from is prsnt, 5], 15]. horm 3.4 constant Stoks Dirac structur: Dnot by R d a compact st and by F = F s, F r, F d a spac of vctor valus smooth functions on, th spac of flows. For simplicity, suppos that E = E s, E r, E d F is th spac of fforts. Morovr, assum that J, G r and G d ar constant matrix diffrntial oprator such that J : E s F s and J = J, G r : F r F s and G d : F d F s. hn, D := { f,, w F E W f s = J s G r f r G d f d r = G r s d = G d s w = B s, f r, f d } is a Stoks Dirac structur with rspct to th pairing f 1, 1, w 1, f,, w {J, Gr, G d } := := 1 f + ] f 1 dv + B {J, Gr, G d }w 1, w da whr B is th analogous of th boundary oprator of Prop. 3.3 and B {J,Gr,G d } is th boundary diffrntial oprator inducd by J, G r and G d on. Proof: Considr f i, i F E, i = 1,. hn, 1 f + ] f 1 dv = s,1 J s, + ] s,j s,1 dv s,1 G r f r, f r,g r s,1 dv s, G r f r,1 f r,1g r s, dv s,1 G d f d, f d,g d s,1 dv s, G d f d,1 f d,1g d s, dv From Lmma 3.1 and its Corollary 3., all th quantitis undr intgration can b xprssd in divrgnc form, that is as th divrgnc of som diffrntial form which is nondgnrat. In particular, dnot by B J, B Gr, B G r, B Gd and B G d th diffrntial oprators inducd on by J, G r and G d and thir adjoint. hn, 1 f + ] f 1 dv = B J s,1, s, da { BGr s,1, f r, + B G r f r,1, s, } da { BGd s,1, f d, + B G d f d,1, s, } da If w i = s,i, f r,i, f d,i, i = 1,, and B B{J, i J i BG i G r, G d } = r B i G d BG i r B i G d with i = 1,..., d, thn it is possibl to writ that 1 f + ] f 1 dv + B {J, Gr, G d }w 1, w da = which, bsid providing th xprssion 18 of th pairing, {J, Gr, G d }, provs that th st dfind in 17 is Stoks Dirac structur on F with rspct to th bilinar form 18. Not 3.6: h prvious thorm is th gnralization of th rsult prsntd in Prop..1 to th constant infinit dimnsional cas. It is possibl, vntually, to introduc th dpndnc on th nrgy variabls and thir spatial drivativs in th diffrntial oprators J, G r and G d. h rsult is th dfinition of nonlinar and stat modulatd Dirac structur in infinit dimnsions. h way in which this rsult can b obtaind rlis on th gnralization to th nonlinar cas of matrix diffrntial oprator and, in particular, of th rsult xprssd by Lmma 3.1. his rsult still holds in th nonlinar cas, but thr ar no rsults concrning th proprtis of th boundary diffrntial oprator B L, in particular about its non dgnracy proprty. Not 3.7: Suppos that f,, w D. From 18, w hav that s f s = r f r dv + d f d dv B {J, Gr, G d }w 1, w da his rlation, which is a dirct consqunc of th dfinition of Dirac structur, xprsss th proprty that th variation of intrnal nrgy is qual to th sum of th dissipatd powr with th powr providd to th systm through th domain and th boundary. IV. MULI-VARIABLE INFINIE DIMENSIONAL POR HAMILONIAN SYSEMS A. Gnral dfinition As in finit dimnsions, th dynamics of a distributd paramtr systm can b obtaind from its Stoks Dirac structur onc th powr ports ar trminatd on th corrsponding lmnts, that is th input/output bhavior of th componnts ar spcifid. Dnot by X th spac of smooth ral valud functions on, + rprsnting th spac of nrgy configuration. h total nrgy is a functional H : X R such that Hx = Hz, x dv whr H is th nrgy dnsity. As proposd in 15], th port bhavior of th nrgy storing lmnt is givn by f s = t s = δ x H

6 whr δ x H is th variational drivativ of th Hamiltonian with rspct to th nrgy configuration. Linar dissipation can b introducd by imposing that f r = Y r r, with r Y r r dv 1 whr Y r : E r F r is a linar oprator. If B is th boundary oprator introducd in 17, from w hav that B s, f r, f d = B s, Y r G r s, f d =: B s, f d and thn th boundary trms can b computd as w = B s, f d. Consquntly, taking into account 17,, 1 and, th following dfinition maks sns. Dfinition 4.1 mdph systm: Dnot by X th spac of vctor valu smooth functions on, + nrgy configurations, by F d th spac of vctor valu smooth functions on distributd flows and assum that E d F d is its dual distributd fforts and by W th spac of vctor valu smooth functions on rprsnting th boundary trms. Morovr, dnot by J a skw-adjoint diffrntial oprator, by G d a diffrntial oprator and by B th boundary oprator dfind in. If H : X R is th Hamiltonian function, th gnral formulation of a multivariabl distributd port Hamiltonian systm with constant Stoks Dirac structur is = J R δ x H + G d f d t d = G d δ xh 3 w = B δ x H, f d whr R := G r Y r G r is a diffrntial oprator taking into account nrgy dissipation and f d, d F d E d. Not 4.1: It is important to not that thr is no a priori distinction btwn flows and fforts in th boundary trms w. hs variabls rsult from th rstriction on of th variational drivativ of H and of its spatial drivativs and, consquntly, thy ar not charactrizd by an xplicit physical maning. In othr words, givn a gnric multivariabl distributd port Hamiltonian systm, th classical structur of powr port, i.. a coupl of signals flow and ffort whos combination givs th powr flow, has bn lost on th boundary. Only if th boundary oprator B {J, Gr, G d } has a particular structur, it is possibl to split th boundary variabl w into two componnts, that is into a flow and an ffort. Proposition 4.1: Considr th mdph systm 3. hn, th following nrgy balanc inquality holds: dh dt = δ x H R δ x H dv + d f d dv + 1 B {J, Gr, G d }w, w da 4 d f d dv + 1 B {J, Gr, G d }w, w da Proof: From, w hav that s f s dv = δ x H dh dv = t dt hn, 4 is immdiat from 19 and 1. Not 4.: Rlation 4 xprsss an obvious proprty of physical systms, that is th variation of intrnal nrgy is lss or qual if no dissipation is prsnt to th powr providd to th systm. In th cas of distributd paramtr systm, th powr can flow insid th systm ithr through th boundary and/or th spatial domain. A. Harry Dym quation V. SIMPLE EXAMPLES h Harry Dym quation is t = 3 3 x 1/ 5 Dnot by =, 1] th spatial domain and by X = L, + th spac of nrgy configurations. h diffrntial oprator J = 3 is skw-adjoint and, thn, it 3 is possibl to dfin a Stoks Dirac structur basd on J as discussd in Prop W giv th following proposition. Proposition 5.1: Dnot by =, 1] th spatial domain and by F = L th spac of flows and assum that E F is th spac of fforts. hn D HD := { f,, w F E W f = 3 z w = B =, z, z } is a Stoks Dirac structur with rspct to th pairing with f 1, 1, w 1, f,, w HD := := 1 1 f + f 1 ] dz + w 1 B J w 1 B J = and W = R 3. Proof: Sinc 3 is a skw-adjoint diffrntial oprator, from Prop. Prop. 3.3 w dduc that it can dfin a Stoks Dirac structur. hn, it is ncssary only to comput B and B J. Givn f i, i F E, i = 1,, w hav that 1 f + f 1 = = which givs B and B J thus concluding th proof. h mdph formulation of th Harry Dym quation is compltd onc th Hamiltonian function is spcifid. In this cas, w hav that Hx := 1 x 1/ z dz

7 thn 5 can b obtaind if, as in, w assum that f = ẋ and = δ x H = x 1/. Clarly, th following nrgy balanc rlation holds: dh dt = δ x H δ x H 1 ] 1 δx H Not that, in this cas, it is not possibl to dfin a pair of flow and ffort variabls on th boundary of th spatial domain s Not 4.1 and that th modl is nonlinar. B. Hat quation h on-dimnsional hat quation is t = x 6 his systm is not Hamiltonian in th classical sns 4], but it can b writtn in mdph form. Dnot by =, 1] th spatial domain and by X = L, + th spac of nrgy configurations. h diffrntial oprator R = is not skw-adjoint and, thn, it is not possibl to rfr to th rsult of Prop. 3.3 in ordr to dfin a Stoks Dirac structur. Dfin th nrgy H of th systm as and thn dh dt = = x Hx = 1 1 xẋ dz = x x dz x dz L x z dz 7 dz 8 his rlation can b intrprtd as an nrgy balanc quation: th variation of intrnal nrgy is lss or qual to th powr providd to th systm through th boundary. In this way, th diffusion phnomnum modld by 6 can b dscribd as pur dissipation. Clarly, a mdph formulation of 6 is possibl only onc a propr Stoks Dirac structur is dtrmind. W giv th following proposition: Proposition 5.: Dnot by =, 1] th spatial domain and by F = L th spac of flows and suppos that E F is th spac of fforts. hn, th st D H := { f s, f r, s, r, w F E W f s = z f r r = z s w = s, f r } 9 is a Stoks Dirac structur on F with rspct to th pairing f 1, 1, w 1, f,, w H = ] = 1 f + ] f 1 dz + w w 1 3 whr W = R. Proof: h proof can b found in 15] sinc 9 is th sam Stoks Dirac structur of th tlgraphr quation or, quivalntly, it can b dducd from horm 3.4 if J =, G r = z and G d =. h hat quation 6 can b obtaind from th Stoks Dirac structur by imposing that f s = ẋ and s = δ x H = x, whr th Hamiltonian function is givn in 7. Morovr, it is ncssary to proprly trminat th rsistiv port f r, r in 9 by supposing that f r = r Finally, th nrgy balanc rlation 8 can b obtaind from 3 sinc givn f s, f r, s, r ; w = ẋ, z δ x H, δ x H, z δ x H; δ x H, z δ x H D H, thn f s, f r, s, r ; w, f s, f r, s, r ; w H =. VI. CONCLUSIONS In this papr, th classical finit dimnsional port Hamiltonian formulation of a dynamical systm is gnralizd in ordr to cop with th distributd paramtr and multivariabl cas and som nw rsults concrning modling and control of distributd paramtr systms in port Hamiltonian form hav bn prsntd. In this way, th dscription of svral physical phnomna, such as hat conduction, is now possibl within this nw port-basd framwork. h cntral rsult is th gnralization of th notion of finit dimnsional Dirac structur to th distributd paramtr cas in ordr to dal with an infinit dimnsional spac of powr variabls. REFERENCES 1]. J. Courant. Dirac manifolds. rans. Amrican Math. Soc. 319, pags , 199. ] M. Dalsmo and A. J. van dr Schaft. On rprsntation and intgrability of mathmatical structurs in nrgy-consrving physical systms. SIAM J. Control and Optimization, 37:54 91, ] G. Golo, V. alasila, and A. J. van dr Schaft. A Hamiltonian formulation of th imoshnko bam modl. In Proc. of Mchatronics. Univrsity of wnt, Jun. 4] A. Gombroff and S. A. Hojman. Non-standard construction of Hamiltonian structurs. J. Phys., A3: , ] R. S. Ingardn and A. Jamiolkowsky. Classical Elctrodynamics. PWN-Polish Sc. Publ., Warszawa, Elsvir, ] A. Macchlli and C. Mlchiorri. Modling and control of th imoshnko bam. th distributd port Hamiltonian approach. Accptd for publication on th SIAM Journal on Control and Optimization, 3. 7] B. M. Maschk and A. J. van dr Schaft. Port controlld Hamiltonian systms: modling origins and systm thortic proprtis. In Procdings of th third Confrnc on nonlinar control systms NOLCOS, ] B. M. Maschk and A. J. van dr Schaft. Intrconnction of systms: th ntwork paradigm. In Proc. 35rd IEEE Conf. on Dcision and Control, pags 7 1, Dc ] B. M. Maschk and A. J. van dr Schaft. Fluid dynamical systms as Hamiltonian boundary control systms. In Proc. of th 4th IEEE Confrnc on Dcision and Control, volum 5, pags , 1. 1] P. J. Olvr. Application of Li groups to diffrntial quations. Springr Vrlag, ] H. M. Payntr. Analysis and dsign of nginring systms. h M.I.. Prss, Cambridg, Massachustts, 1961.

8 1] M. Rnardy and R. C. Rogrs. An Introduction to Partial Diffrntial Equations. Numbr 13 in xts in Applid Mathmatics. Springr Vrlag, nd dition, 4. 13] G. E. Swatrs. Introduction to Hamiltonian fluid dynamics and stability thory. Chapman & Hall / CRC,. 14] A. J. van dr Schaft. L -Gain and Passivity chniqus in Nonlinar Control. Communication and Control Enginring. Springr Vrlag,. 15] A. J. van dr Schaft and B. M. Maschk. Hamiltonian formulation of distributd-paramtr systms with boundary nrgy flow. Journal of Gomtry and Physics,.